Understanding Theile/Small Parameters
This mathematical description of loudspeaker behavior applies to any sound transducer, but finds it's most common application with enclosure design for woofers/subwoofers. For most audiophiles, a standard enclosure that meets the VAS requirement for the their driver will usually perform quite adequately. The three Theile/Small parameters that primarily determine the frequency response of a loudspeaker are:
Compliance or Vas
Free-air resonance or Fs
Response time or Qts
Compliance or Vas is a measure of the overall stiffness, or resistance to motion of the cone, the surround, and the spider. It is specified in term of the volume of air having the same compliance as the driver. A small number corresponds to a small volume of air, which is stiffer than a larger volume of air. Thus, compliance and stiffness are inversely proportional.
Optimum enclosure volume is proportional to Vas. It is usually measured in cubic feet, and generally is equivalent to the internal volume of the enclosure. This is the principle specification for the speaker cabinet.
Free-air resonance or Fs is the resonant frequency of the driver's voice coil impedance with the driver suspended in free air (no enclosure). The -3 dB frequency (F3) of an enclosure is proportional to Fs.
Response time or Qts is a measure of the sharpness of the driver's free-air resonance. It is defined as (Fh-Fl)/Fs, where Fh and Fl are the upper and lower -3 dB points of the driver's voice coil impedance in free air. Optimum enclosure volume is related to Qts but is not directly proportional. It is accurate to say that the volume gets larger as Qts gets larger. Likewise, F3 gets smaller as Qts gets larger, and for the sealed box enclosure, F3 is inversely proportional to Qts.
Theile/Small Parameters And Speaker Design
The reference nomenclatures for the Physical Characteristics of a speaker are:
Re: The static D.C. resistance of the voice coil measured in Ohms.
Sd: The surface area of the speaker's cone.
BL: The magnetic strength of the motor structure.
Mms: The total moving mass of the speaker including the small amount of air in front of and behind the cone.
Cms: The stiffness of the driver's suspension.
Rms: The losses due to the suspension.
The Theile/Small Response Parameters
By understanding the relationship of these 20 physical parameters and how to change them, we may alter the response of the system to fit the desired goal.
The most frequently cited measures in design are:
Re: The D.C. resistance of the voice coil measured in Ohms.
Sd: The surface area of the speaker.
Fs: The resonant frequency of the speaker.
Q: The ratio of resistance to reactance, or vice versa.
Qes: The electrical "Q" of the speaker.
Qms: The mechanical "Q" of the speaker.
Qts: The total "Q" of the speaker.
Vas: The volume of air having the same acoustic compliance as the speaker's suspension.
Xmax: The width of the voice coil that extends beyond the front plate plus 15%. This relates to how far the speaker can move in either direction without appreciable distortion. The amount of power required to move a speaker to its maximum excursion is referred to as the displacement limited power handling. Please note that this number varies with enclosure size and type.
Understanding the response parameters allows us to calculate the predicted frequency response of a given speaker system. The formulas that accomplish this are rather lengthy and complex, and are best left to a computer. There are a number of high quality computer programs on the market that automate the design process of building an enclosure.